Is the COVID-19 pandemic curve a Gaussian curve? We've all heard a lot about "flattening the curve". I was wondering if these curve – that look like bells – can be qualified as Gaussian despite the fact that there is a temporal dimension.

 A: No.
For example:


*

*Not in the sense of a Gaussian probability distribution: the bell-curve of a normal (Gaussian) distribution is a histogram (a map of probability density against values of a single variable), but the curves you quote are (as you note) a map of the values of one variable (new cases) against a second variable (time). (@Accumulation and @TobyBartels point out that Gaussian curves are mathematical constructs that may be unrelated to probability distributions; given that you are asking this question on the statistics SE, I assumed that addressing the Gaussian distribution was an important part of answering the question.)

*The possible values under a normal distribution extend from $-\infty$ to $\infty$, but an epidemic curve cannot have negative values on the y axis, and traveling far enough left or right on the x axis, you will run out of cases altogether, either because the disease is does not exist, or because Homo sapiens does not exist.

*Normal distributions are continuous, but the phenomena epidemic curves measure are actually discrete not continuous: they represent new cases during each discrete unit of time. While we can subdivide time into smaller meaningful units (to a degree), we eventually run into the fact that individuals with new infections are count data (discrete).

*Normal distributions are symmetric about their mean, but despite the cartoon conveying a useful public health message about the need to flatten the curve, actual epidemic curves are frequently skewed to the right, with long thin tails as shown below.



*

*Normal distributions are unimodal, but actual epidemic curves may feature one or more bumps (i.e. may be multi-modal, they may even, as in @SextusEmpiricus' answer, be endemic where they return cyclically).

*Finally, here is an epidemic curve for COVID-19 in China, you can see that the curve generally diverges from the Gaussian curve (of course there are issues with the reliability of the data, given than many cases were not counted):

A: The most simple analysis of an epidemic leads to a logistics curve model. The rate of new infections will be the derivative of total cases, which under that model would give a bell-shaped curve (normal-ish in the middle but with much fatter tails -- see Dirk's comment below).
The assumptions behind the model are a constant rate of transmission, exactly as would be the case for exponential growth, but unlike exponential growth there is the presence of a saturation limit. In many epidemics the saturation limit would be the entire population (i.e. eventually everyone will have been exposed and acquired immunity). In the case of COVID-19 that's hopefully not going to be the case so some hand-wavy adjustment will be needed such that the spread limits at some sub-set of the whole population.
My source for this was this excellent youtube video 1. (Maybe there is some better source than youtube?)
A: I'm no epidemiologist myself, but another key difference between that curve and a Gaussian curve is that the Gaussian decays to zero relatively fast (as $e^{-t^2}$ after some time $t$), while an actual epidemic can be expected to taper off at a much slower rate at the end, or might even not decay to $0$ but to some other (hopefully low) constant – i.e. the virus might not die out entirely like the Gaussian curve suggests.
A: Epidemiological curves for respiratory infections are very irregular curves. See for instance the SARS outbreak of 2002/2003

https://www.who.int/csr/sars/epicurve/epiindex/en/index1.html
and for endemic diseases they may have some seasonal pattern. See for instance the euromomo logo

(source: euromomo.eu)
Besides the flattening the curve in general not being a Gaussian curve, the situation will also be more nuanced. The image that goes around on the internet is a very extreme case were the curve sticks a lot above the threshold and is being halved in size as result of the measures. It sketched a perfect situation to argue for drastic measures. That may not necessarily be so much the case with covid-19.
More nuanced representations show different thresholds and have more subtle differences in the curves. Like here

https://www.vaccinarsinpuglia.org/notizie/2017/10/al-via-la-sorveglianza-dellinfluenza-stagione-2017-18
A: I'm not an epidemiologist, and you should ask this question to the epidemiologists. 
First of all, drawing Gaussian curves is simple, since even basic plotting software has them implemented (e.g. Microsoft Excel), so when people need to draw "a distribution", they often draw Gaussians. The "flatten the curve" figures are aimed to show the general idea of the phenomenon, not the exact distribution of that will and could have happen (nobody knows it in advance, since there is too many unknowns, and too many moving parts). Even the scales of the figures are not realistic; some experts point that the difference may be much higher than on such figures.
As about Gaussian shape of the epidemic, as far as I know, this is known as Farr's law. First the number of infected people rises, then falls, so this is similar to a Gaussian curve, but it is far from an exact fit. You can find discussion in this Twitter thread, that gives as an example of a study that applied Farr's law to predicting HIV/AIDS cases in US, as you can see from the plot, it has nothing to do with the actual outcome. 

You can find some, more serious, figures in the widely cited recently paper by Ferguson et al (2020). As you can see, they are "rising and falling, but far from Gaussian, in some simulations even multimodal, or skewed. Of course, this is still a simulation, so a much more simplified distribution than what we could expect from actual data.


A: It seems like there are three questions here:


*

*Is the actual distribution of cases Gaussian? No. 

*Are the curves given in the graphic Gaussian? Not quite. I think the red one is a little bit skewed, and the blue one is definitely skewed.

*Can plots of a value versus time be considered Gaussian? Yes. 

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the form $$f(x) = ae^{-{\frac {(x-b)^{2}}{2c^{2}}}}$$
  for arbitrary real constants a, b and non zero c. 

https://en.wikipedia.org/wiki/Gaussian_function
There is no requirement that it be a probability distribution.
A: Not but (under the right assumptions that in practice aren't likely to hold) sort of.
As Michael Reid points the number of infected people of an epidemic under simplified constant conditions (constant R0) is governed by the logistic equation, which leads to a sigmoid, the logistic function. The derivative of the logistic function is the bell shaped density curve of the logistic distribution, which is not normal in spite of looking normal at first glance. Since the derivative would represent number of new infected people per time unit and common metrics like the number of deaths per day or the number newly reported cases per day are more or less proportional to a delayed and unfocused version of the number of new infected people, they also follow a curve similar to the logistic distribution density function.
However, some assumptions of the logistic equation may not hold for the coronavirus outbreak - in fact, they may not hold for any real population, although the logistic equation is a common and useful model in population dynamics:


*

*In the dynamic equation it is assumed that all the population reproduces, that is that all the people that has been infected keeps infecting more people. In reality, at some point infected people stop spreading the infection.

*Conditions (R0) are assumed constant. In real world, contention measures are introduced and therefore R0 changes.

A: Short answer, no. I was wondering the same thing and I found out a way to plot populations of susceptible, infected, and recovered people.  It's a model called a compartmental model of epidemiology and the specific algorithm is called the Gillespie Algorithm.  There's Python code in the second link but I tried it in R and it looks like this  and here's the notebook if you're interested.  
It seems like something like Poisson distribution would be closer, but under the right conditions, we could approximate the Poisson with a normal/Gaussian distribution. That's the generous interpretation. The other interpretations are: 1, the CDC actually doesn't know the right shape, or 2, the CDC wants to dumb it down for public consumption.
A: In fact, this curve seems to fit well an Inverse Gausssian distribution. This distribution is widely used in psychology or economics for describing the distribution of time delays. Indeed, there are similarities of such processes with a pandemic (where what is denoted in the graph by the variable $x$ will be time since the start of the pandemic):

The source code for applying such a fitting procedure to your own data is available in this notebook
Note that for certain values, this curve may look close to a Gaussian "bell-shaped" distribution. The mean and standard deviation control the time of the peak and the "spread" the curve. Still, the fitting error will be less using the inverse Gaussian distribution. Knowing how the precision of inferred parameters are on generic political decisions and the final fatality rate, the choice of a fitting procedure must be accurately validated.
A: No. As demonstrated here on various countries, so far, a reasonable way to model the curves of daily new confirmed cases and deaths for Covid-19 is to use:


*

*an increasing exponential at the very beginning

*a logistic curve when the curve starts to flatten (see 3Blue1Brown's video) 

*an decreasing exponential shortly after the first peak

*afterwards, we might lack data to tell.


See for example Italy as of the 22nd of April 2020 (with Logistics fit before peak, exponential after):

As for the USA, the logistics model is enough so far:

Finally, it is harder to tell for China:

A: In the early stages of an epidemic growth is exponential. The two key parameters are R0 (average number of people infected by each person who catches it) and incubation time. The goal is to reduce R0 - once it is less than 1.0 the epidemic is over. Most counties are still at that stage for COVID-19.
Once a significant fraction of the population becomes immune, an exponential model is no longer a good fit. See user953847's great answer above. 
Sextus Empiricus points out that actual data are irregular. That is true of any real data. Nevertheless, ideal models can be useful as a way to find and communicate trends underlying the irregularities.
A: Biological growth (cumulative) of virus epidemics, or trees, or humans, or other biological phenomena, in general follows the logistic function: 1/(1+e^-1). The logistic curve is sigmoid or S-shaped. It does not "flatten" but it has an inflection point.
The first derivative is the growth rate. That curve follows the logistic distribution. It is bell-shaped like the Gaussian curve, although it is different. F(x) = e^-x/(1+e^-x)^2. The peak of the growth rate curve is contemporaneous (because the x-axis is time) with the inflection point of the cumulative growth curve.
The second derivative is acceleration. It is S-shaped on its side, like a sine wave skewed to the right. Acceleration passes through the x-axis (equals zero) when the rate peaks and cumulative growth inflects. Thereafter acceleration is negative (deceleration) and after dipping into negative territory it asymptotically approaches the x-axis from below.
The Gompertz function is a specialized case of the general logistic function, and is sometimes used for growth studies because it has parameters that can be solved for via linear regression. One of the parameters is the upper asymptote of the cumulative growth curve. That parameter would correspond to total deaths or total cases if those were what you were estimating.
Also sometimes used is the Weibull distribution, another specialized case with parameters. We used the Weibull to develop so-called individual tree stand growth models back when I was a grad student.
That is the math of growth. It is not "exponential" or "logarithmic". It is logistic.
